MinT - Best Known (24−23, 24, s)-Nets in Base 25

Best Known (24−23, 24, s)-Nets in Base 25

(24−23, 24, 27)-Net over F₂₅ — Constructive and digital

Digital (1, 24, 27)-net over F₂₅, using

net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
- Niederreiter sequence [i]

(24−23, 24, 36)-Net over F₂₅ — Digital

Digital (1, 24, 36)-net over F₂₅, using

net from sequence [i] based on digital (1, 35)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 1 and N(F) ≥ 36, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(24−23, 24, 113)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (1, 24, 114)-net over F₂₅, because

extracting embedded orthogonal array [i] would yield linear OA(25²⁴, 114, F₂₅, 23) (dual of [114, 90, 24]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(25²³, 27, F₂₅, 23) (dual of [27, 4, 24]-code or 27-arc in PG(22,25)), but
    - bound for MDS codes [i]
  2. OA(25⁹⁰, 114, S₂₅, 87), but
    - discarding factors would yield OA(25⁹⁰, 96, S₂₅, 87), but
      - the linear programming bound shows that MÂ â‰¥ 13 668865 221078 739256 015071 809629 879850 936124 576679 322503 501034 318504 064092 312472 780759 593604 486823 448240 784273 366443 812847 137451 171875 / 20 559539 > 25⁹⁰ [i]

(24−23, 24, 123)-Net in Base 25 — Upper bound on s

There is no (1, 24, 124)-net in base 25, because

15 times m-reduction [i] would yield (1, 9, 124)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁹, 124, S₂₅, 8), but
  - the linear programming bound shows that MÂ â‰¥ 119725 208199 920654 296875 / 31015 449154 > 25⁹ [i]